\section{Standard Errors By Portfolio Size}
\label{sec:StandardErrorsByPortfolioSize}

When insurers A, B, PI, D and E randomly select portfolios of: 307,000,000; 10,000,000; 1,000,000; 100,000 and 10,000 policyholders from the same population, they select PLREs from different, normally distributed, populations. Each insurer's distribution is centered at the PLR, but the standard errors for portfolios of size "n," are calculated using our assumption that s1,000,000 = 0.05 and this formula: 
	
\begin{equation}
s_n = s_{1,000,000} * \frac{1,000,000}{\sqrt{n}}	
\end{equation}
	
The greater accuracy of larger insurer's PLREs, and the lower accuracy of smaller insurer's PLREs, are reflected in the sizes of their standard errors: 0.0029; 0.0158; 0.0500; 0.1581 and 0.5000, respectively, as shown in Exhibit 1 Row 4.

Although the loss ratios we used earlier were 0, 1, 2, 3 and 4 standard errors above the PLR for PI, these same loss ratio evaluation points will be higher (lower) numbers, of standard error units, above the PLR for insurers larger (smaller) than PI, affecting their respective cumulative probabilities and the probabilities assigned to specific operating results. Larger insurers have more probability of loss ratios below evaluation points higher than the PLR, and small insurers have more probability of loss ratios above all evaluation points higher than the PLR. As a result, larger insurers are more likely to earn profits and avoid losses than smaller insurers, despite the fact that all insurers select individual policyholders, at random, from the same population.
